Splitting methods in the design of coupled flow and mechanics simulators

Transcription

1 Splitting methods in the design of coupled flow and mechanics simulators Sílvia Barbeiro CMUC, Department of Mathematics, University of Coimbra PhD Program in Mathematics Coimbra, November 17, 2010 Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 1 / 36

2 Introduction Modeling porous media systems aims to describe the changes in a permeable media (made of somewhat rigid material containing open spaces) due to changes of the fluid it held. Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 2 / 36

3 Introduction The changes in pressure caused by motion, removal or addition of liquid or gas may cause deformations in the structure holding the fluid. In the past, modelers tended to ignore the geomechanics in their calculations. Side effects of drilling: consolidation - reduction in volume due to fluid extraction compaction - reduction in volume due to air removal subsidence - collapse Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 3 / 36

4 Motivation Subsidence of Ekofisk Oil Field was a side effect of drilling. Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 4 / 36

5 Motivation They subsided so much they had to go in and raise the platforms, costing them several billion dollars. If they d known ahead of time, they could have built their platforms taller Rick Dean, in Modeling complex, multiphase porous media systems, Siam News, April Photo: Norwegian Petroleum Museum/ConocoPhillips Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 5 / 36

6 Motivation Subsidence of Venice cased by the extraction of the aquifer. Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 6 / 36

7 Modeling Poroelasticity refers to fluid flow within a deformable porous medium under the assumption of relatively small deformations. Examples of poroelastic structures soil, rock, cartilage, brain, heart, bone Various environmental, energy industry and biomechanics applications Subsidence, reservoir compaction Well stability, sand production Waste disposal Sequestration of carbon in saline aquifers Estimate tumor-induced stress levels in the brain Development of prosthetic devices for cartilage, bone, heart valves Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 7 / 36

8 Biot s consolidation model Karl von Terzaghi (October 2, October 25, 1963) Austrian civil engineer and geologist. Frequently called the father of soil mechanics. Maurice Anthony Biot (May 25, September 12, 1985) Belgian-American physicist Founder of the theory of poroelasticity. Karl von Terzaghi Maurice Anthony Biot Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 8 / 36

9 Biot s consolidation model primary variables: displacement u = (u 1, u 2, u 3 ) and fluid pressure p Balance of linear momentum (σ(u) αpi ) = f σ(u) - effective stress tensor, linear elastic, f - body force, α - Biot-Willis constant where σ(u) = 2µɛ(u) + λtr(ɛ(u))i, ɛ(u) = 1 2 ( grad u + (grad u) t), λ, µ - Lamé constants The momentum conservation equation is very similar to the equation governing linear elasticity, the exception is the addition of the term involving pressure. Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 9 / 36

10 Biot s consolidation model Mass conservation η - fluid content, v f - flux of fluid, s f η t = v f + s f - volumetric fluid source term Equation for fluid content η in terms of fluid pressure p and material volume u η = c 0 p + α u c 0 - constrained specific storage coefficient Darcy s law for the flux of fluid v f = 1 µ f K( p ρ f g) K - permeability tensor, µ f - fluid viscosity, g - gravity Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 10 / 36

11 Summary of equations Coupling equations In the domain, at any time t Initial conditions at time t = 0 (σ(u) αpi ) = f t (c 0p + α u) 1 µ f K( p ρ f g) = s f Plus adequate boundary conditions p(0) = p 0, u(0) = u 0 For the mixed formulation for the flow, we introduce the variable z = 1 µ f K( p ρ f g). Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 11 / 36

12 Variational problem Integrating by parts over Ω, we obtain the variational problem Find u, p and z such that a u (u, v) α(p, v) = ( p ) ( ) c 0 t, w + α t u, w Ω f v + r N v, Γ N s f w, Ω p D s η Γ p + ( z, w) = (µ f K 1 z, s) (p, s) = ρ f g s holds for all (v, w, s) and t [0, T ], where a u (u, v) = (2µ(ɛ(u) : ɛ(v)) + λ( u)( v)) dx. Ω Ω Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 12 / 36

13 Time discretization Using the backward Euler method. Let n denote the time step, and t the time increment a u (u n+1, v) α(p n+1, v) = ( ) ( c 0 (p n+1 p n ), w + α (µ f K 1 z n+1, s) (p n+1, s) = holds for all (v, w, s). Ω f v + r N v, Γ N ) + t( z n+1, w) = t ρ f g s p D s η Ω Γ p (u n+1 u n ), w Ω s f w, Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 13 / 36

14 Time discretization Operator splitting α u = c r p + c r α σ FLOW ( ) (c 0 + c r )(p n+1,k+1 p n ), w + t( z n+1,k+1, w) = t s f w Ω ) α ( σn+1,k σ n ), w, ( cr (µ f K 1 z n+1,k+1, s) (p n+1,k+1, s) = p D s η + Γ p MECHANICS a u (u n+1,k+1, v) = Ω f v + r N v + α(p n+1,k+1, v) Γ N Ω ρ f g s Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 14 / 36

15 Splitting: iterative coupling Time step loop Time iteration n = n + 1 k = k + 1 FLOW σ n,k 1 MECHANICS No Converges? Yes Convergence criterion σ n+1,k σ n+1,k 1 L < Tol Iterative coupling is stable and accurate. [Wheeler and Gai, Numer. Meth. PDEs, 2007] Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 15 / 36

16 Fully, iteratively, explicit and loosely coupled Fully coupled The coupled governing equations of flow and geomechanics are solved simultaneously at every time step. Iteratively coupled Either the flow, or mechanical, problem is solved first, then the other problem is solved using the intermediate solution information. This sequential procedure is iterated at each time step until the solution converges to within an acceptable tolerance. The converged solution is identical to that obtained using the fully coupled approach. Explicitly coupled This is a special case of the iteratively coupled method, where only one iteration is taken. Loosely coupled The coupling between the two problems is resolved only after a certain number of flow time steps. This method can save computational cost compared to the other strategies, but it is less accurate and requires reliable estimates of when to update the mechanical response. [Kim, Tchelepi, Juanes, SPE, 2009] Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 16 / 36

17 Space discretization A very simple example u (x) + u(x) = (1 + π 2 ) sin(πx), x (0, 1), u(0) = 0, u(1) = 0 Variational formulation Multiplying the equation by any arbitrary weight function v H0 1 (0, 1) and integrating over the interval (0, 1) 1 0 u (x)v(x) dx + Integrating by parts 1 0 u(x)v(x) dx = 1 0 (1 + π 2 ) sin(πx)v(x) dx. 1 u (x)v (x) dx u(x)v(x) dx = 1 0 (1 + π 2 ) sin(πx)v(x) dx +u (1)v(1) u (0)v(0). Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 17 / 36

18 Finite element method The finite element method supplies an approximation to the analytical solution in the form of a piecewise polynomial function, defined over the entire computational domain. Example: The simplest case of linear splines. 1 ϕ i (x x i 1 )/h i, x i 1 x x i, ϕ i (x) = (x i+1 x)/h i+1, x i x x i+1, 0, otherwise. x i 1 x i x i+1 x A piecewise linear finite element basis function ϕ i (hat functions). Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 18 / 36

19 Finite element method x 0 = a x 1 x 2 x 3 x 4 x 5 x 6 x 7... A piecewise linear function. x n = b x Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 19 / 36

20 Finite element method Matrix form Since the aim of Finite Element Method method is the production of a linear system of equations, we build its matrix form, which can be used to compute the solution by a computer program. n We expand u n in respect to this basis, u n = c j ϕ j to obtain j=1 n c j a(ϕ j, ϕ i ) = f (ϕ i ) i = 1,..., n, j=1 which is a linear system of equations AU = F, where a ij = a(ϕ j, ϕ i ), F i = f (ϕ i ). Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 20 / 36

21 Finite element method For the finite element method the important property of the basis functions ϕ i, 1 i n is that they have local support, being nonzero only in one pair of adjacent intervals (x i 1, x i ] and [x i+1, x i ). This means that, A ij = 0 if i j > 0. = The matrix A is symmetric, positive definite and tridiagonal, and the associated system of linear equations can be solved very efficiently. Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 21 / 36

22 Finite element method Back to the very simple example (Exact solution: u(x) = sin(πx)) Uniform subdivision of [0, 1] of spacing h = 1/n. n = 2 n = 4 Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 22 / 36

23 Finite element method Uniform subdivision of [0, 1] of spacing h = 1/n. n = 6 n = 100 In the last figure, the approximation error is so small that u and u n are indistinguishable. Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 23 / 36

24 Finite element method Basis function in 2D Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 24 / 36

25 Finite element method Find u n V n such that v n V n, a(u n, v n ) = f (v n ). Galerkin orthogonality The key property of the Galerkin approach is that the error is orthogonal to the chosen subspaces. Since V n V, we can use v n as a test vector in the original equation. Subtracting the two, we get the Galerkin orthogonality relation for the error, e n = u u n and a(e n, v n ) = a(u, v n ) a(u n, v n ) = f (v n ) f (v n ) = 0 a(e n, e n ) = a(u u n, u u n ) = min v n V n a(u v n, u v n ). Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 25 / 36

26 Finite element method Galerkin orthogonality u ϕ H0 1 (a, b) u u n = (u ϕ) (u n ϕ) V n 0 u n ϕ Energy norm: v E = a(v, v) 1/2 u n is the best approximation from V n to the weak solution u H0 1 (a, b), when we measure the error of the approximation in the energy norm. Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 26 / 36

27 Spatial discretization E h and E H be two nondegenerate partitions of the polyhedral domain Ω with maximal element diameter h and H, respectively. Mixed spaces for flow variables Examples of mixed spaces with the needed properties are the Raviart-Thomas-Nedelec spaces. Example: Lowest order Raviart-Thomas 2D triangles: in each element p =const, z = (a + bx, c + by) t quadrilaterals: in each element p =const, z = (a + bx, c + dy) t Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 27 / 36

28 Mandel s problem Mandel s solution has been used as a benchmark problem for testing the validity of numerical codes of poroelasticity. [Mandel, 1953] - analytical solution for pressure [Abousleiman et al., 1996] - analytical solution for displacement and stress 2F y x 2F Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 28 / 36

29 Mandel s problem (λ + µ) ( u) µ 2 u + α p = 0 in Ω (0, T ] t (c 0p + α u) 1 µ f K p = 0 in Ω (0, T ] boundary conditions p = 0, x = a, 1 µ f K p η = 0, x = 0, y = 0, y = b, u 1 = 0, x = 0, u 2 = 0, y = 0, u y = 0, y = b, x ση = ( F /a)η, y = b, ση = 0, x = 0, x = a, y = 0, u x = 0 z x = 0 u y =0 z y =0 p = 0 computational domain Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 29 / 36

30 Mandel s problem surface: p, arrows: u 1 and u 2 T = Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 30 / 36

31 Mandel s problem surface: p, arrows: u 1 and u 2 T = 0.1 Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 31 / 36

32 Mandel s problem surface: p, arrows: u 1 and u 2 T = 0.2 Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 32 / 36

33 Mandel s problem surface: p, arrows: u 1 and u 2 T = 0.5 Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 33 / 36

34 Mandel s problem Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 34 / 36

35 Mandel s problem Numerical results H u ū H 1 rate p p L 2 rate z z L 2 rate 5.000e e e e e e e e e e e e e e e e e e e e e e e e-2 - Convergence rates: bilinear elements for u, lowest order Raviart-Thomas space for p and z Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 35 / 36

36 Important Questions Is the method stable? Is the method convergente? Next seminar... I really enjoy developing efficient and accurate solutions to real-world problems, while maintaining a solid theoretical base. MFW Sílvia Barbeiro (CMUC) Splitting methods for flow and mechanics simulators 36 / 36